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Fix waveguide eigenvalue derivation

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Thanks to Rafael Diaz Fuentes and Paolo Pintus for catching and
correcting these!
This commit is contained in:
jan 2021-07-05 15:01:08 -07:00
parent 23490694ff
commit fcca9e3ae5
1 changed files with 45 additions and 14 deletions
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57
meanas/fdfd/waveguide_2d.py
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@ -69,8 +69,8 @@ $$
- \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (-\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z) \\\\
&= \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x ( \\imath \\omega \\epsilon_{xx} E_x)
- \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (-\\imath \\omega \\epsilon_{yy} E_y) \\\\
\\gamma \\tilde{\\partial}_x E_z &= \\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
\\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y) \\\\
\\gamma \\tilde{\\partial}_x E_z &= \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y) \\\\
\\end{aligned}
$$
@ -78,23 +78,32 @@ With a similar approach (but using $\\gamma \\tilde{\\partial}_y$ instead), we c
$$
\\begin{aligned}
\\gamma \\tilde{\\partial}_y E_z &= \\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
\\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y) \\\\
\\gamma \\tilde{\\partial}_y E_z &= \\tilde{\\partial}_y \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\tilde{\\partial}_y \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y) \\\\
\\end{aligned}
$$
We can combine this equation for $\\gamma \\tilde{\\partial}_y E_z$ with
the unused $\\imath \\omega \\mu_{xx} H_z$ and $\\imath \\omega \\mu_{yy} H_y$ equations to get
the unused $\\imath \\omega \\mu_{xx} H_x$ and $\\imath \\omega \\mu_{yy} H_y$ equations to get
$$
\\begin{aligned}
-\\imath \\omega \\mu_{xx} \\gamma H_x &= \\gamma^2 E_y + \\gamma \\tilde{\\partial}_y E_z \\\\
-\\imath \\omega \\mu_{xx} \\gamma H_x &= \\gamma^2 E_y + \\tilde{\\partial}_y (
\\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
) \\\\
\\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
)\\\\
\\end{aligned}
$$
and
$$
\\begin{aligned}
-\\imath \\omega \\mu_{yy} \\gamma H_y &= -\\gamma^2 E_x - \\gamma \\tilde{\\partial}_x E_z \\\\
-\\imath \\omega \\mu_{yy} \\gamma H_y &= -\\gamma^2 E_x - \\tilde{\\partial}_x (
\\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
\\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
)\\\\
\\end{aligned}
$$
@ -106,10 +115,10 @@ $$
\\begin{aligned}
-\\imath \\omega \\mu_{xx} (\\gamma H_x) &= -\\imath \\omega \\mu_{xx} (-\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z) \\\\
&= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y
-\\imath \\omega \\mu_{xx} \\hat{\\partial}_x (
+\\imath \\omega \\mu_{xx} \\hat{\\partial}_x (
\\frac{1}{-\\imath \\omega \\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x)) \\\\
&= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y
+\\mu_{xx} \\hat{\\partial}_x \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
-\\mu_{xx} \\hat{\\partial}_x \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
\\end{aligned}
$$
@ -117,12 +126,30 @@ and, similarly,
$$
\\begin{aligned}
-\\imath \\omega \\mu_{yy} (\\gamma H_y) &= -\\omega^2 \\mu_{yy} \\epsilon_{xx} E_x
-\\imath \\omega \\mu_{yy} (\\gamma H_y) &= \\omega^2 \\mu_{yy} \\epsilon_{xx} E_x
+\\mu_{yy} \\hat{\\partial}_y \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
\\end{aligned}
$$
By combining both pairs of expressions, we get
$$
\\begin{aligned}
-\\gamma^2 E_x - \\tilde{\\partial}_x (
\\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
) &= \\omega^2 \\mu_{yy} \\epsilon_{xx} E_x
+\\mu_{yy} \\hat{\\partial}_y \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
\\gamma^2 E_y + \\tilde{\\partial}_y (
\\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
) &= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y
-\\mu_{xx} \\hat{\\partial}_x \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
\\end{aligned}
$$
Using these, we can construct the eigenvalue problem
$$ \\beta^2 \\begin{bmatrix} E_x \\\\
E_y \\end{bmatrix} =
(\\omega^2 \\begin{bmatrix} \\mu_{yy} \\epsilon_{xx} & 0 \\\\
@ -137,6 +164,10 @@ $$ \\beta^2 \\begin{bmatrix} E_x \\\\
E_y \\end{bmatrix}
$$
where $\\gamma = \\imath\\beta$. In the literature, $\\beta$ is usually used to denote
the lossless/real part of the propagation constant, but in `meanas` it is allowed to
be complex.
An equivalent eigenvalue problem can be formed using the $H_x$ and $H_y$ fields, if those are more convenient.
Note that $E_z$ was never discretized, so $\\gamma$ and $\\beta$ will need adjustment